Nonlinear fermions and coherent states

Abstract

Nonlinear fermions of degree n (n-fermions) are introduced as particles with creation and annihilation operators obeying the simple nonlinear anticommutation relation AA † + A † n A n = 1. The (n + 1)-order nilpotency of these operators follows from the existence of unique A-vacuum. Supposing appropreate (n + 1)-order nilpotent paraGrassmann variables and integration rules the sets of n-fermion number states, 'right' and 'left' ladder operator coherent states (CS) and displacement-operator-like CS are constructed. The (n + 1) × (n + 1) matrix realization of the related para-Grassmann algebra is provided. General (n+1)-order nilpotent ladder operators of finite dimensional systems are expressed as polynomials in terms of n-fermion operators. Overcomplete sets of (normalized) 'right' and 'left' eigenstates of such general ladder operators are constructed and their properties briefly discussed.

Extended pseudo-fermions from non commutative bosons

“…For this reason, we call these particles, generalized pseudo-fermions. Other interesting generalizations of fermions could be found, for instance, in [17,18] and in [19].…”

Generalized Grassmann variables for quantum kit (k-level) systems and Barut–Girardello coherent states for su(r + 1) algebras

“…We hope that this novel construction of generalized Grassmann variables for qukit systems and Barut-Girardello coherent states for su(r + 1) algebras will be of interest in the field of quantum systems with finite dimensional Hilbert space , especially for pseudo-Hermitian quantum systems. Also, it is interesting to investigate the relation between the generalized Weyl-Heisenberg algebras discussed in this paper and the formalism of nonlinear fermions discussed in [47].…”

Nonlinearn-pseudo fermions

“…So the set of |ψ k is a basis in H n+1 , but since b = a † this basis is not orthogonal. One can check (using the anticommutation relation (16)) that the states |ψ k are eigenstates of the non-Hermitian operator N pf (n) = b(n)a(n) + b 2 (n)a 2 (n) + . .…”